- Final Exam Review will
take place on Monday, December 14, 10-12 in BA2179.
- The Final will take place on
**Tuesday, December 15, 10-1 in BA6183****. **
- It will cover everything we've
**covered
in class, lecture notes and on homework**.
Specifically, it will cover the following material:

- Chapter 3: Everything except functors.

- Chapter 4: Everything except for covering maps.
- Chapter 5: Submanifolds, embeddings.
- Chapter 6: Sard's Theorem, Whitney embedding
theorem for compact manifolds, Transversality.

- Chapter 8 : Vector fields, criteria of
smoothness, sufficient condition for the triviality of
the tangent bundle.
- Chapter 9: Integral curves and integral flows.

- Chapter 12: Everything except tensor products of
vector spaces.
- Chapter 14: Alternating tensors, cotangent
bundle, tensor bundles on manifolds, differential forms,
exterior derivatives.
- Chapter 15: Orientations on vector spaces,
orientations on manifolds.
- Chapter 16: Integration of forms, Stokes's
Formula, applications to De Rham cohomology, Integration
on Riemannian manifolds

- Chapter 17: Mayer-Vietoris sequence, Cohomology
with compact support, degree theory.
- Euler charactersitic of orientable manifolds and
Euler characteristic mod 2.

- Rules for the Final :
**No
aids
allowed.** No
calculators or notes! You can quote results from the
book, lectures and homework.

- Here are some practice problems
on recent material to help you prepare for the Final.
I also recommend that you do the following problems from the
book:
- 1-7, 1-11, 2-6, 2-8, 3-1, 3-6, 3-8, 4-2,
4-7, 5-1, 5-6, 5-7, 6-2, 6-9, 8-3, 8-4, 8-11, 9-3d, 9-4,
9-5, 14-7ab, 15-13a